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Growth of hypercyclic functions: a continuous path between $\mathcal{U}$-frequent hypercyclicity and hypercyclicity

Part of: Special classes of linear operators Miscellaneous topics of analysis in the complex domain General theory of linear operators

Published online by Cambridge University Press:  08 May 2024

Augustin Mouze  and
Vincent Munnier
Augustin Mouze*
Affiliation:
CNRS, UMR 8524 - Laboratoire Paul Painlevé, École Centrale de Lille, Univ. Lille, Lille, France
Vincent Munnier
Affiliation:
Lycée Jacques Prévert, Boulogne Billancourt, France
*
Corresponding author: Augustin Mouze, email: augustin.mouze@univ-lille.fr
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Abstract

We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$-frequent hypercyclicity and hypercyclicity.

Keywords

hypercyclic vectorsweighted shiftsrate of growthboundary behaviour

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 30E25: Boundary value problems 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B38: Operators on function spaces (general)

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 3 , August 2024 , pp. 794 - 829
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Bayart, F., Grivaux, S., Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 11 (2006), 5083–5117.CrossRefGoogle Scholar
Bayart, F., Matheron, E., Dynamics of Linear Operators, Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2000.Google Scholar
Bernal-González, L., Bonilla, A., Rate of growth of hypercyclic and frequently hypercyclic functions for the Dunkl operator, Mediterr. J. Math. 13 5 (2016), 3359–3372.CrossRefGoogle Scholar
Bernal-González, L., Universal functions for Taylor shifts, Complex Var. Theory Appl. 31 2 (1996), 121–129.Google Scholar
Bès, J., Menet, Q., Peris, A., Puig, Y., Recurrence properties of hypercyclic operators, Math. Ann. 366 1–2 (2016), 545–572.CrossRefGoogle Scholar
Blasco, O., Bonilla, A., Grosse-Erdmann, K-G., Rate of growth of frequently hypercyclic functions, Proc. Edinb. Math. Soc. 53 1 (2010), 39–59.CrossRefGoogle Scholar
Bonilla, A., Grosse-Erdmann, K-G., Upper frequent hypercyclicity and related notions, Rev. Mat. Complut. 31 3 (2018), 673–711.CrossRefGoogle Scholar
Drasin, D., Saksman, E., Optimal growth of frequently hypercyclic entire functions, J. Funct. Anal. 263 11 (2012), 3674–3688.CrossRefGoogle Scholar
Duren, P.L., Theory of p Spaces, Pure and Applied Mathematics, 38, Academic Press, New-York-London, 1970.Google Scholar
Edwards, R.E., Gaudry, G.I., Littlewood-Paley and Multiplier Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90, Springer-Verlag, Berlin-New York, 1977.CrossRefGoogle Scholar
Ernst, R., Esser, C., Menet, Q., $\mathcal{U}$-frequent hypercyclicity notions and related weighted densities, Israel J. Math. 241 2 (2021), 817–848.CrossRefGoogle Scholar
Ernst, R., Mouze, A., A quantitative interpretation of the frequent hypercyclicity criterion, Ergodic Theory Dynam. Systems 39 4 (2019), 898–924.CrossRefGoogle Scholar
Ernst, R., Mouze, A., Frequent universality criterion and densities, Ergodic Theory Dynam. Systems 41 3 (2021), 846–868.CrossRefGoogle Scholar
Freedman, A. R., Sember, J. J., Densities and summability, Pacific J. Math. 95 2 (1981), 293–305.CrossRefGoogle Scholar
Gethner, R. M., Shapiro, J. H., Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 2 (1987), 281–288.CrossRefGoogle Scholar
Gilmore, C., Saksman, E., Tylli, H-O., Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator, Proc. Roy. Soc. Edinburgh Sect. A 149 6 (2019), 1577–1594.CrossRefGoogle Scholar
Grosse-Erdmann, K-G., Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987).Google Scholar
Grosse-Erdmann, K-G., On the universal functions of G. R. MacLane, Complex Variables Theory Appl. 15 3 (1990), 193–196.Google Scholar
Grosse-Erdmann, K-G., Rate of growth of hypercyclic entire functions, Indag. Math. (N.S.) 11 4 (2000), 561–571.Google Scholar
Grosse-Erdmann, K-G., Hypercyclic and chaotic weighted shifts, Studia Math. 139 1 (2000), 47–68.CrossRefGoogle Scholar
Grosse-Erdmann, K-G., Peris, A., Linear Chaos, Universitext, Springer, London, 2011.CrossRefGoogle Scholar
Kitai, C., Invariant Closed Sets for Linear Operators, Ph.D. thesis, Univ. Toronto, 1982.Google Scholar
Menet, Q., A bridge between $\mathcal{U}$-frequent hypercyclicity and frequent hypercyclicity, J. Math. Anal. Appl. 482 2 (2020), .CrossRefGoogle Scholar
Mouze, A., Munnier, V., Frequent hypercyclicity of random holomorphic functions for Taylor shifts and optimal growth, J. Anal. Math. 143 2 (2021),615–637.CrossRefGoogle Scholar
Mouze, A., Munnier, V., Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc, Canad. Math. Bull. 64 2 (2021), 264–281.CrossRefGoogle Scholar
Shkarin, S., On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc. 137 (2009), 123–134.CrossRefGoogle Scholar
Rudin, W., Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10 (1959), 855–859.CrossRefGoogle Scholar